Korteweg-de Vries equation family in the theory of nonlinear internal waves

As it is known, the canonical Korteweg-de Vries equation is applied to describe nonlinear long internal waves in the first approximation on parameters of nonlinearity and dispersion. To compare with surface gravity waves, the coefficient of quadratic nonlinearity can have either sign and to be zero. In this case, the asymptotic procedure should take into account higher terms of nonlinearity. Generalized Korteweg-de Vries equation called the Gardner equation is now a popular model to analyze nonlinear internal waves in the ocean with complicated density and shear flow stratification. If the density stratification is almost linear, the number of nonlinear terms is increased. The family of the Korteweg-de Vries-like equations for internal waves in the form u_t+ [F(u)]_x + u_xxx = 0 is discussed in this presentation. In leading order the nonlinear term is F(u) ~ qu^b  with b > 0. The steady-state travelling solitary waves is analyzed.

            For q > 0 and b > 1 the analysis re-confirmed that all travelling solitons have “light” exponentially decaying tails and propagate to the right. If q < 0 and b < 1, the travelling solitons (so called compactons) have a compact support (and thus vanishing tails) and propagate to the left. For more complicated F(u) and b > 1 (e.g., the Gardner equation and higher-order generalizations) standing algebraic solitons with “heavy” power-law tails may appear. If the leading term of F(u) is negative, the set of solutions may include wide or table-top solitons (similar to the solutions of the Gardner equation), including algebraic solitons and compactons with any of the three types of tails. The solutions usually have a single-hump structure but if F(u) represents a higher-order polynomial, the generalized KdV equation may support multi-humped pyramidal solitons.

Study is supported by RFBR Grant No 21-55-15008.